Question

select the correct answer from each drop-down menu. andy is designing a dice tray in the shape of a rectangular prism to use during a role-playing game. the tray needs to be three centimeters high and have a volume of 252 cubic centimeters in order for the dice to roll properly. the length of the tray should be five centimeters longer than its width. the volume of a rectangular prism is found using the formula v = l · w · h, where l is the length, w is the width, and h is the height. complete the equation that models the volume of the tray in terms of its width, x, in centimeters. drop-down x² + drop-down x = drop-down is it possible for the width of the tray to be 7.5 centimeters? drop-down reset next

Explanation:

Answer:

First, let's define the variables:

• Width: \( x \) (in cm)
• Length: \( x + 5 \) (since it's 5 cm longer than the width)
• Height: \( 3 \) cm (given)
• Volume: \( V = 252 \) cubic cm

The volume of a rectangular prism is \( V = l \cdot w \cdot h \). Substituting the values:

\( 252 = (x + 5) \cdot x \cdot 3 \)

Simplify the equation:

Divide both sides by 3: \( 84 = x(x + 5) \)

Expand the right side: \( 84 = x^2 + 5x \)

Rearrange to standard quadratic form: \( x^2 + 5x - 84 = 0 \)

Now, let's check if the width can be 7.5 cm. Substitute \( x = 7.5 \) into the volume formula:

Length: \( 7.5 + 5 = 12.5 \) cm

Volume: \( 12.5 \cdot 7.5 \cdot 3 = 12.5 \cdot 22.5 = 281.25 \) cubic cm, which is not equal to 252. So, let's solve the quadratic equation \( x^2 + 5x - 84 = 0 \)

Factor the quadratic: \( (x + 12)(x - 7) = 0 \)

So, \( x = -12 \) (discard, since width can't be negative) or \( x = 7 \)

Now, for the first part (completing the equation):

The equation is \( x^2 + 5x - 84 = 0 \) (so the blanks would be +5x, -84, =0)

For the second part (is the width 7.5 cm possible?): No, because when we solve the equation, the width is 7 cm, and substituting 7.5 gives a different volume.

So, putting it all together:

First drop - down (coefficient of x): 5

Second drop - down (constant term): -84

Third drop - down (right - hand side): 0

For the "Is it possible..." part: No

So the answers are:

First blank: 5

Second blank: -84

Third blank: 0

Is it possible: No